We prove a new version of the quantum accuracy threshold theorem that applies
to non-Markovian noise with algebraically decaying spatial correlations. We
consider noise in a quantum computer arising from a perturbation that acts
collectively on pairs of qubits and on the environment, and we show that an
arbitrarily long quantum computation can be executed with high reliability in D
spatial dimensions, if the perturbation is sufficiently weak and decays with
the distance r between the qubits faster than 1/r^D.Comment: 4 page

Alexei Kitaev

D. Aharonov

Dorit Aharonov

John Preskill

P. W. Shor

English

arXiv

We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1/r^D

Aharonov, Dorit

Kitaev, Alexei

Preskill, John

Caltech Authors - Main

PRL 96, 050504 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 FEBRUARY 2006Fault-Tolerant Quantum Computation with Long-Range Correlated Noise
Dorit Aharonov,1 Alexei Kitaev,2,3 and John Preskill2
1School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel
2Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA
3Microsoft Research, One Microsoft Way, Redmond, Washington 98052, USA
(Received 31 October 2005; published 7 February 2006)0031-9007=We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian
noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising
from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an
arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the
perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1=rD.
DOI: 10.1103/PhysRevLett.96.050504 PACS numbers: 03.67.PpThe theory of quantum fault tolerance [1] establishes
that quantum information, if cleverly encoded, can be
protected from damage and processed reliably with imper-
fect equipment. One noteworthy achievement of this the-
ory is the quantum threshold theorem [2–6], which asserts
that an arbitrarily long quantum computation can be exe-
cuted with high reliability, provided that the noise afflicting
the computer’s hardware is weaker than a certain critical
value, the accuracy threshold.
Early proofs of the threshold theorem [2– 4] assumed
that the noise is Markovian, that each quantum gate in the
noisy circuit is a trace-preserving completely positive map
that approximates the ideal gate. Each gate in this noise
model can be realized as a unitary transformation that acts
jointly on a set of the qubits in the computer (which we call
system qubits) and on the environment (the bath variables),
but where it is assumed that the bath has no memory—the
state of the bath is refreshed after every gate. In [7] the
theorem was extended to a non-Markovian noise model of
a restricted type. It was assumed that each qubit Qi has a
local bath Ei, and that Ei and Ej interact only when a
quantum gate acts on the corresponding pair of qubits. In
[5], the noise model was relaxed further by allowing non-
local interactions among the bath variables, but it was still
assumed that a pair of system qubits interact directly only
when a quantum gate acts on that pair. In this Letter, we
show that the threshold theorem applies even if there are
long-range interactions among the system qubits that are
‘‘always on.’’
The noise model we consider can be formulated in terms
of a time-dependent Hamiltonian H that governs the joint
evolution of the system and the bath. We may expressH as
H  HS HB HSB; (1)
where HS is the time-dependent Hamiltonian of the system
that realizes the ideal quantum circuit, HB is an arbitrary
Hamiltonian of the bath, andHSB couples the system to the
bath. In [5] it was assumed that the system-bath coupling
could be expressed as06=96(5)=050504(4)$23.00 05050HSB 
X
a
HSB;a; (2)
here a labels a circuit ‘‘location’’ during a particular time
step of the computation—either a single qubit at a location
where a one-qubit gate acts in the ideal circuit (which
could be the identity gate in the case of a ‘‘resting’’ qubit),
or a pair of qubits at a location where a two-qubit gate acts
on the pair in the ideal circuit. This coupling of the system
qubits to one another is ‘‘short range’’ in the sense that two
system qubits interact directly with one another only if the
ideal Hamiltonian HS also couples those two data qubits
during the same time step—hence we refer to it as the
short-range noise model. For short-range noise it was
shown in [5] that there is a positive constant "0 such that
an arbitrarily long ideal quantum computation can be
accurately simulated using the noisy gates (with reasonable
overhead) provided that at each time and at each location
kHSB;akt0 < "0; (3)
where t0 is the time needed to execute a quantum gate, and
the sup operator norm k  k is defined by
kAk  supj i0 kAj ikk j ik :
In this Letter we consider a system-bath coupling of the
form
HSB 
X
hiji
Hij; (4)
where the sum is over all pairs of system qubits; Hij acts
collectively on the pair of qubits hiji and on the bath. (By
convention, Hij  Hji  Hhiji.) Since any pair of system
qubits can interact, we call this the long-range noise model.
We show that for long-range noise there is a positive
constant 0 such that an arbitrarily long ideal quantum
computation can be accurately and efficiently simulated
provided that for each qubit i and at each time4-1 © 2006 The American Physical Society
PRL 96, 050504 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 FEBRUARY 2006
X
j
kHijkt0 <0: (5)
We note that if the interaction between system qubits
decays algebraically so that
kHijk< =ji jjz; (6)
where ji jj denotes the distance between qubits i and j,
then the sum in Eq. (5) converges in the limit of an infinite
D-dimensional system provided that z > D. Thus, our
result shows that fault-tolerant quantum computation is
possible for  sufficiently small.
As in [5,7], we analyze the noisy circuit using a ‘‘fine-
grained fault-path expansion.’’ We divide the working
period t0 of a gate into t0=  1 intervals each of width
; for short-range noise the time evolution operator for the
time interval t; t can then be expressed as
Ut; t  eiH  eiHSeiHBY
a
ISB  iHSB;a
(7)
since for  sufficiently small terms higher order in  can
be safely neglected. The action Ucircuit on the system and
bath of a noisy quantum circuit with altogether N time
steps can be expressed as a product of Nt0= such evolu-
tion operators. The fine-grained fault-path expansion of the
noisy circuit is constructed by expressing Ucircuit as a sum
of monomials, where in each term either ISB or iHSB;a
is inserted at each ‘‘microlocation’’ with duration . A
microlocation where iHSB;a is inserted is called a ‘‘mi-
crofault.’’ A noisy gate or ‘‘macrolocation’’ in the circuit
consists of t0= consecutive microlocations, and the mac-
rolocation simulates the ideal gate perfectly if it contains
no microfaults. Therefore, we say that a macrolocation is
faulty (is a ‘‘macrofault’’) if it contains one or more micro-
faults. By grouping together terms in the fine-grained fault-
path expansion, we obtain a coarse-grained fault-path ex-
pansion such that each term in the coarse-grained expan-
sion is the sum of all terms in the fine-grained expansion
with macrofaults occurring at a specified set of
macrolocations.
The strength of the noise in the circuit can be quantified
by the norm of the sum EI1 over all the terms in the fine-
grained expansion such that a particular specified macro-
location I1 is faulty (where no restriction is placed on the
noise at other macrolocations). We can organize this sum
by noting that there must be an earliest microfault in the
macrolocation, which can occur at any of the t0= micro-
locations; summing over all fine-grained fault paths with a
fixed earliest microfault is equivalent to inserting ISB at the
microlocations prior to the earliest microfault and inserting
eiHSB;a at the later microlocations. Since keiHSB;ak  1
and the norm of a product is bounded above by a product of
norms, the norm of the sum over all fine-grained paths with
a fixed earliest microfault is no larger than kHSB;ak. Now
summing over the t0= possible microlocations for the05050earliest microfault, and noting that the norm of a sum is
bounded above by a sum of norms, we have
kEI1k 	 " 
 MaxkHSB;akt0; (8)
where the maximum is with respect to all a and all times.
By similar reasoning, if EI r denotes the sum over all
fine-grained fault paths such that faults occur at a particular
set I r of r macrolocations (where no restriction is placed
on the noise at other macrolocations), then
kEI rk 	 "r: (9)
If Eq. (9) is satisfied, we say that the noise is ‘‘local with
strength ".’’ Note that " is determined only by the system-
bath coupling HSB; no restriction need be imposed on the
bath Hamiltonian HB.
Thus we see that the short-range noise model obeys the
local noise condition Eq. (9). For a quantum computer
subjected to local noise, the effectiveness of a recursive
fault-tolerant simulation of an ideal circuit was analyzed in
[5]. In a level-1 simulation, each gate of an ideal circuit is
replaced by a 1-gadget, a circuit that simulates the gate
acting on quantum information protected by a quantum
error-correcting code. In a level-k simulation, each gate of
an ideal circuit is replaced by a k-gadget, which is con-
structed by replacing each gate in a k 1-gadget by a 1-
gadget. If the code can correct t errors, then a 1-gadget is
said to be bad if it contains t 1 or more faulty macro-
locations, and a k-gadget is said to be bad if it contains t
1 or more bad k 1 locations.
If the 1-gadgets are properly constructed, then it can be
shown that for a level-1 simulation, the norm of the sum
EI 1r  over fault paths such that the 1-gadgets are bad at a
specified set I 1r of r level-1 locations satisfies
kEI 1r k 	 "1r; (10)
where
"1  C

A
t 1

"t1; (11)
here A is the maximal number of macrolocations contained
in any 1-gadget, and C is a constant of order 1.
Furthermore, by an inductive argument we find that for a
level-k simulation, the norm of the sum EI kr  over fault
paths such that the k-gadgets are bad at a specified set I kr
of r level-k locations satisfies
kEI kr k 	 "kr; (12)
where
"k  C

A
t 1

"k1t1  "0"="0t1k ; (13)
and4-2
PRL 96, 050504 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 FEBRUARY 2006"0 

C

A
t 1
1=t
: (14)
Thus, for " < "0, the norm drops steeply as the level k
increases, and it can be shown that an ideal circuit with L
gates can be simulated with fixed accuracy and an over-
head cost that is polylogarithmic in L. This is the quantum
threshold theorem for local noise [5].
To establish a threshold theorem for the long-range noise
model, then, it suffices to show that the local noise condi-
tion Eq. (9) holds for the long-range noise, just as it does
for short-range noise. For now, suppose that all of the gates
performed at the r specified faulty macrolocations are
single-qubit gates. If these macrolocations all occur at
different times, then our earlier analysis of short-range
noise needs little modification. Now we express the time
evolution operator for time interval t; t  as
Ut ; t  eiHSeiHBY
hiji
I  iHij (15)
and again expand Ucircuit to generate a fine-grained fault-
path expansion. Considering a fixed faulty macrolocation
I1 where the ideal gate acts on qubit i, again there must be
an earliest microfault that acts on qubit i in that macro-
location, which can be at any of the t0= intervals; we
insert iPjHij at that earliest microfault, I at each
earlier microlocation, and eiHSB at all later time inter-
vals. Then summing over the position of the earliest micro-
fault we find
kEI1k 	  
 Max
X
j
kHijk

t0; (16)
where the maximum is with respect to all i and all times.
Similarly, if macrofaults occur at a set I r of r macro-
locations, where all the macrolocations are at different
time steps, then by summing independently over the ear-
liest microfault at each macrolocation, we find
k EI r k	 r: (17)
But what if many macrofaults occur during the same
working period? The long-range noise model allows faults
occurring in the same time step to be correlated. The worst
case is when the r specified faulty macrolocations in the set
I r are all in the same time step. In that case, if i and j are
two of the qubits acted upon by the r faults, then Hij can
account for two of the faults.
We again organize the sum over fine-grained fault paths
by considering the earliest microfault that strikes each of
the r qubits. For a particular fine-grained fault path, if
i; j 2 I r and Hij is the first perturbation to act on both
qubits i and j, then we say that the pair ij is ‘‘contracted.’’
If Hij is the first perturbation to act on qubit i, where either
j =2 I r or j 2 I r has already been struck earlier by another
perturbation, then we say that qubit i is ‘‘uncontracted.’’ By
definition, then, the contracted pairs are disjoint.05050We group the fine-grained fault paths into families,
where in each family the time of the earliest microfault
acting on each qubit in I r is specified, as well as the pair of
qubits acted upon by that fault. It is clear that the norm of
the sum of fine-grained fault paths in a family can be
bounded by a product—with a factor of kHijk arising
from a fault that acts on the pair hiji; the only problem is to
count the families.
In each family, a specified set of k pairs are contracted,
where k ranges from 0 to r=2 for r even, or from 0 to r
1=2 for r odd. Each contraction can occur in any of the
t0= time steps, which converts the factor of  to t0. For an
uncontracted qubit i, we sum over the qubit j it could be
paired with by the perturbation Hij—the sum runs over
both j 2 I r and j =2 I r. For j =2 I r, this microfault can
occur at any time, but for j 2 I r that might not be true. If
we slide Hij forward in time to before the first microfault
acting on qubit i, then we are considering a family in which
i and j are contracted. But that is all right; if we allow Hij
to occur at any time, we are only overestimating the norm
of the sum over fault paths; so here again we have an upper
bound by converting  to t0, and summing over j gives a
factor of  for each uncontracted qubit.
Let us suppose that r  2n is even. The norm Sk of the
sum over fine-grained fault paths with k contractions can
be bounded above:
Sk 	
X
contractions
2n2k
Yk
‘1
kHi‘;j‘kt0
	 1
2kk!
X
i;j
kHijkt0

k
2n2k 	 1
2kk!
2nk2n2k:
(18)
(Note that the sum of kHijk over all i and j, raised to the
power k, contains a term bounding the contribution from
each contraction 2kk! times, and also contains other posi-
tive terms.) Thus, we have
Sk 	 1k=ek n
k2nk  ek

n
k

k
2nk: (19)
To bound the sum over k we use

n
k

k 

k
n
k=nn 	 e1=en (20)
and n 1 	 en to obtain
kEI rk
Xn
k0
Sk	n1enen=en	e21=en0r;
(21)
where
0  e11=2e p : (22)
Thus, we have shown that, even when all of the r  2n4-3
PRL 96, 050504 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 FEBRUARY 2006
specified faults occur during the same working period, the
norm of the sum over fault paths is bounded by 0r, where
0  O p . This bound also applies when r is odd.
The same bound applies to two-qubit gates, except that
we need to replace  by 2. A two-qubit gate has a fault if
the perturbation hits either one of the two qubits (and after
one of the qubits is damaged, we do not care what happens
to the other one). The first perturbation to damage the pair
of qubits i and j might act on the pair im (m  j), on the
pair mj (m  i), or on the pair ij. So the norm of the sum
over fault paths that damage one particular two-qubit
macrolocation can be bounded above by
X
mj
kHimk 
X
mi
kHmjk  kHijk

t0 	 2: (23)
When we consider summing over the fault paths with
many faults in the same time step, where some of the faults
occur at two-qubit macrolocations, now the uncontracted
contributions include the case where a single perturbation
damages both of the qubits at a single macrolocation;
if, on the other hand, a two-qubit macrolocation is con-
tracted with another macrolocation, the perturbation can
act on either qubit at the two-qubit macrolocation. Both
types of contributions are taken into account by replacing
 by 2.
The conclusion is that ‘‘long-range noise’’ is really a
type of ‘‘local noise’’ obeying Eq. (9), where
"  e11=2e 2p ; (24)
therefore the quantum accuracy threshold theorem proven
in [5] is applicable. Note that we can use similar reasoning
for many-body interactions. For example, if the system-
bath interaction is a sum of three-body terms, then
 

Max
X
jk
kHijkk

t0; (25)
and "  O1=3.
Actually, even the ‘‘short-range’’ noise model includes
long-range spatial correlations among the faults. If each
system qubit couples individually to the bath with strength
, but the self-interactions of the bath may be nonlocal,
then a coupling between a pair of system qubits of order 2
can be induced in second-order perturbation theory, and
p-body terms of order p are generated in higher order.
Nevertheless, the ‘‘long-range’’ noise model is more gen-
eral, because there are interactions among the system
qubits that are allowed in the long-range model but cannot
be simulated by a local coupling of the system to the bath.
Furthermore, by working directly with a system-bath
Hamiltonian that includes long-range interactions among05050the system qubits, we can see more easily how the rate of
decay of the interactions with distance enters into the
analysis, as in Eq. (6).
Using methods described in [5], we can obtain a lower
bound on the threshold noise strength "0  105 or 0 
1010 for the case of two-body interactions. This estimate
is discouraging, but it is probably much too pessimistic for
several reasons. In the analysis we allow the bad fault paths
to add together with a common phase, so that they inter-
fere constructively. Under a more plausible scenario in
which the phases of distinct fault paths are only weakly
correlated, the norm of the sum is significantly smaller,
so that much stronger noise can be tolerated in practice.
Furthermore, by following sound engineering principles,
for example, by spatially separating qubits belonging to the
same code block, we can improve the effectiveness of
fault-tolerant protocols. We also note that more clever
gadget design could significantly improve estimates of
the threshold as suggested, for example, in [8], and in
any case the actual value of the threshold is likely to be
notably higher than can be established by rigorous argu-
ments. The noteworthy point is that we have shown that
sufficiently weak long-range interactions among the qubits
(and interactions with a nonlocal bath that can store infor-
mation for an indefinitely long time) are compatible in
principle with fault-tolerant quantum computation.
We thank Daniel Gottesman for helpful comments. This
work has been supported in part by DOE under Grant
No. DE-FG03-92-ER40701, NSF under Grant No. PHY-
0456720, ARO under Grants No. W911NF-04-1-0236,
No. W911NF-05-1-0294, and No. DAAD19-00-1-0374,
ISF under Grants No. 032-9739 and No. 039-7549, the
U.S. Army under Grant No. 030-7657, and the Council of
Higher Education in Israel under Grant No. 033-7233.4-4[1] P. W. Shor, in Proceedings of the 37th Annual Symposium
on Foundations of Computer Science (IEEE Press, Los
Alamitos, CA, 1996), pp. 56–65.
[2] D. Aharonov and M. Ben-Or, in Proceedings of the 29th
Annual ACM Symposium on Theory of Computing (ACM,
New York, 1997), p. 176; D. Aharonov and M. Ben-Or,
quant-ph/9906129.
[3] A. Yu. Kitaev, Russ. Math. Surv. 52, 1191 (1997).
[4] E Knill, R. Laflamme, and W. H. Zurek, Proc. R. Soc. A
454, 365 (1998).
[5] P. Aliferis, D. Gottesman, and J. Preskill, quant-ph/
0504218.
[6] B. W. Reichardt, quant-ph/0509203.
[7] B. M. Terhal and G. Burkard, Phys. Rev. A 71, 012336
(2005).
[8] E. Knill, Nature (London) 434, 39 (2005).


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Fault-tolerant quantum computation with long-range correlated noise

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